1. Field of the Invention
The present invention relates generally to ring arithmetic operations and particularly to efficient modular exponentiation of large numbers.
2. Description of Related Art
Modern society has seen information transmission dramatically grow in prevalence, and the importance of information security has likewise grown. Transmitting information over an open network—such as the Internet—involves many security challenges.
The most common Internet protocol for transmitting secured information is Transport Layer Security (TLS), descendent of Secure Sockets Layer (SSL). For clarity and because of the protocols' similarities, reference will be made to SSL/TLS throughout this application. To improve speed, SSL/TLS uses symmetric encryption to encrypt much of the transmitted data. But symmetric encryption is vulnerable because communicants must share a private key.
For improved security, SSL/TLS uses the slower asymmetric encryption to share symmetric keys. But every session requires sharing of a new private key because key reuse would substantially increase vulnerability. So in practice new sessions are established frequently, forcing heavy usage of asymmetric encryption.
Some of the principal Internet transactions using this type of security are e-commerce transactions. In a transaction of this type, the consumer transmits identifying information as well as credit-card or other financially sensitive data to a vendor. The amount of data that must be encrypted to complete the transaction is very small, typically less than twenty lines of text. The time spent by a server encrypting this data is insignificant compared with the time necessary to encrypt and decrypt the symmetric key in the asymmetric key-exchange portion of the transaction. Because each session requires a new key, which must be encrypted and then decrypted using the slow asymmetric encryption process, whenever a significant number of sessions are established, the majority of server resources may be dedicated to the key exchange protocol.